Consider the angle pair classifications: adjacent, complementary, supplementary, and vertical angles. Can two angles fit all four categories?

No

Step-by-step explanation:

Take two angles α and β. If I said that angle α and β are complementary angles is correct to say α+β = 90°, because of the definition of complementary angles. The supplementary angles are those that sum 180°, and it is impossible that α+β=90° and at the same time α+β=180°.

Something similar occurs with the adjacent and vertical angles, if they are adjacent, this means they have a common vertex and side but if they are vertical, α is in the opposite position of β and the angles are formed by the intersection of two lines. So, β cannot be at the side of α and at the same time in the opposite position.